Midterm
W1D1: Jan. 4th, 2022
About Class
PHd in Statistics from Idaho
Homework Due on Friday, and will be graded over the weekend.
Starting with Chapter 5.
Class notes are in Canvas
461 Last Lecture Notes
Joint CDF and PDF
CDF for two random variables have 3 properties
smallest value is 0
largest value is 1
always non-decreasing
Joint PMF for discrete variables uses summation and has two properties
all probability are between 0 and 1
total probability must be 1
Example: Two-way Table
Two way tables, because there are two ways to read it
also known as pivot table and contingency table
make sure you can write and understand in different ways
a) P(Only 1 car will be sold)
b) P(1 warrenty will be sold given that 2 cars were sold on a given day)
c) Find and report the marginal pmf of \(Y_1\)
“O otherwise” must be included for a complete distribution function.
d) Are \(Y_1\) and \(Y_2\) independent?
W1D2: Jan. 6th, 2022
Review Homework will be Homework 1 and due Friday 14th
Joint PDF
Properties :
smallest value is 0
total volume under the joint pdf is 1
Notes :
If the limits are given without equal sign then you can apply integral in the same way.
Example : Joint pdf
a) Draw and shade the support
The support is the area on the graph.
Upper limit is only important for PMF of discrete random variables
pdf : \(f_{Y_1,Y_2}\geq0\)
PMF : \(0\leq p(y_1,y_2)\leq 1\)
b) Verify \(f(y_1,y_2)\) is a valid joint pdf
c) Find the joint CDF
CDF : \(F(y_1,y_2)=P(Y_1\leq y_1, Y_2\leq y_2)=\int_{-\infty}^{y_2}\int_{-\infty}^{y_1}f(y_1,y_2)dt_1dt_2\)
when we do problems we do not need to write every step of the integral
pay attention to the limits on the double integral (\(y_1\) is on the inner integral, meaning it is integrated first (\(dt_1\) inside \(dt_2\)))
final answer will be a function, not a limit
lowest value must be 0, and highest value must be 1
- 0 otherwise is wrong without values for 1 included in CDF
when you find the cdf you must use \(dt_1\) and \(dt_2\), but don’t need it to use or solve for probabilities
d) P(Both variables are less than 1/4)
e) P(\(Y_1\) is more than 0.75)
W2D3: Jan. 11th, 2022
W2D4: Jan. 13th, 2022
W3D5: Jan. 18th, 2022
joint pmf (p) : between 0 and 1
joint pdf (f) : nonnegative
Discrete uses summation
W3D6: Jan. 20th, 2022
W4D7: Jan. 25th, 2022
W4D8: Jan. 27th, 2022
W5D9: Feb. 1st, 2022
MIDTERM